Absolute continuity and singularity of two probability measures on a filtered space

Abstract

Let $\mu$ and $\nu$ be fixed probability measures on a filtered space $(\Omega ,\mathcal{F},({\mathcal{F}}_t{)}_{t\in {\mathbf{R}}^{+}})$. Denote by ${\mu }_T$ and ${\nu }_T$ (respectively, ${\mu }_{T-}$ and ${\nu }_{T-}$) the restrictions of the measures $\mu$ and $\nu$ on ${\mathcal{F}}_T$ (respectively, on ${\mathcal{F}}_{T-}$) for a stopping time $T$. We find the Hahn decomposition of ${\mu }_T$ and ${\nu }_T$ using the Hahn decomposition of the measures $\mu$, $\nu$, and the Hellinger process $h_t$ in the strict sense of order 1/2. The norm of the absolutely continuous component of ${\mu }_{T-}$ with respect to ${\nu }_{T-}$ is computed in terms of density processes and Hellinger integrals.

arXiv comment18 pages, no figures